Question

v(x) = x(8 - 2x)(10 - 2x) solve step by step

Answer 1

To solve the equation \( v(x) = x(8 - 2x)(10 - 2x) \), we will start by expanding the equation step by step. This will give us a polynomial form which makes it easier to analyze and solve.

### Step 1: Expand the Expression

First, let's expand \( (8 - 2x)(10 - 2x) \).

\[
(8 - 2x)(10 - 2x) = 8 \cdot 10 - 8 \cdot 2x - 2x \cdot 10 + 2x \cdot 2x
\]

Calculating each term:

\[
8 \cdot 10 = 80
\]
\[
-8 \cdot 2x = -16x
\]
\[
-2x \cdot 10 = -20x
\]
\[
2x \cdot 2x = 4x^2
\]

Now combine these:

\[
(8 - 2x)(10 - 2x) = 80 - 16x - 20x + 4x^2 = 80 - 36x + 4x^2
\]

### Step 2: Substitute Back

Now substitute this back into the original equation for \( v(x) \):

\[
v(x) = x(80 - 36x + 4x^2)
\]

Distributing \( x \):

\[
v(x) = 80x - 36x^2 + 4x^3
\]

### Step 3: Rearrange into Standard Polynomial Form

Rearranging the polynomial gives us:

\[
v(x) = 4x^3 - 36x^2 + 80x
\]

### Step 4: Set the Polynomial Equal to Zero

To solve for \( x \), set \( v(x) = 0 \):

\[
4x^3 - 36x^2 + 80x = 0
\]

### Step 5: Factor Out Common Terms

We can factor out \( 4x \):

\[
4x(x^2 - 9x + 20) = 0
\]

### Step 6: Solve Each Factor

The equation \( 4x = 0 \) gives:

\[
x = 0
\]

Next, we solve the quadratic \( x^2 - 9x + 20 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a = 1, b = -9, c = 20 \):

\[
b^2 - 4ac = (-9)^2 - 4 \cdot 1 \cdot 20 = 81 - 80 = 1
\]

Now plug in the values into the formula:

\[
x = \frac{9 \pm \sqrt{1}}{2}
\]
\[
x = \frac{9 \pm 1}{2}
\]

This gives us two potential solutions:

1. \( x = \frac{10}{2} = 5 \)
2. \( x = \frac{8}{2} = 4 \)

### Step 7: Solutions

Thus, the solutions to the equation \( v(x) = 0 \) are:

\[
x = 0, \quad x = 4, \quad x = 5
\]

These are the points where the polynomial intersects the x-axis or where \( v(x) = 0 \).